2012/01/26 by Brightwell, Graham, Luczak, Malwina
#37A25 #60F15 #60J10 #Combinatorics (math.CO) #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1201.5523
In the supermarket model, there are n queues, each with a single server. Customers arrive in a Poisson process with arrival rate λn, where λ= λ(n) ∈ (0,1). Upon arrival, a customer selects d=d(n) servers uniformly at random, and joins the queue of a least-loaded server amongst those chosen. Service times are independent exponentially distributed random variables with mean~1. In this paper, we analyse the behaviour of the supermarket model in a regime where λ(n) tends to~1, and d(n) tends to infinity, as n → ∞. For suitable triples (n,d,λ), we identify a subset \cal N of the state space where the process remains for a long time in equilibrium. We further show that the process is rapidly mixing when started in \cal N, and give bounds on the speed of mixing for more general initial conditions.